How do you find the area of a circle sector?

Category: 2D Shapes »

Course: Geometry »

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Sector Area

Circle sectors are partial circles. They are often smaller than a semi-circle and look like a "pizza slice" but can be bigger than a semi-circle and look like the remaining pizza after you remove a slice (which I affectionately refer to as a "pac-man").orTheir most important quality is the fact that they are a partial circle. If we know the area of the whole circle, we can find the area of a fraction of it.A sector has the same radius length as the whole circle to which it belongs. We know the area of a circle is $\pi r^2$, so all we have to do is consider what fraction of the circle the sector represents, and then take that same fraction of the total area.The key lies in the central angle. If a sector has a central angle of $x$°, then the sector represents $x/360$ fraction of the whole circle. Indeed, we can see that if the central angle was the full $360$°, that the sector would be one whole circle.Therefore, the area of a sector with radius $r$ and central angle $x$° is given by the formula:$$\mathrm{Area} = \frac{x}{360} \cdot \pi r^2$$which again, should be interpreted as [fraction] $\times$ [whole circle area].If you are measuring your central angle in radian measure, then we will ratio the central angle $\theta$ against $2\pi$ radians, since $360$° is equivalent to $2\pi$ radians.$$\mathrm{Area} = \frac{\theta}{2 \pi} \cdot \pi r^2$$Additionally, as a small shortcut, if you are measuring your central angle in radians, you can simplify the sector area equation to say$$\mathrm{Area} = \frac{1}{2} \cdot r^2 \theta$$where $\theta$ is the central angle of the sector measured in radians.